{"about":{"site":"https://codewithpapers.app","non_affiliation":"Code with Papers and Syntology are not affiliated with, endorsed by, or sponsored by Papers with Code, Meta, or the pwc-archive mirror.","licence":"CC BY-SA 4.0","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","attribution":"https://codewithpapers.app/attribution","modified":"archive material modified by Syntology; see the attribution page"},"url":"/paper/an-optimal-fpt-algorithm-parametrized-by","title":"Solving Cut-Problems in Quadratic Time for Graphs With Bounded Treewidth","arxiv_id":"2101.00694","date":"2021-01-03","proceeding":null,"authors":["Hauke Brinkop","Klaus Jansen"],"abstract":"In the problem (Unweighted) Max-Cut we are given a graph $G = (V,E)$ and asked for a set $S \\subseteq V$ such that the number of edges from $S$ to $V \\setminus S$ is maximal. In this paper we consider an even harder problem: (Weighted) Max-Bisection. Here we are given an undirected graph $G = (V,E)$ and a weight function $w \\colon E \\to \\mathbb Q_{>0}$ and the task is to find a set $S \\subseteq V$ such that (i) the sum of the weights of edges from $S$ is maximal; and (ii) $S$ contains $\\left\\lceil{\\frac{n}{2}}\\right\\rceil$ vertices (where $n = \\lvert V\\rvert$). We design a framework that allows to solve this problem in time $\\mathcal O(2^t n^2)$ if a tree decomposition of width $t$ is given as part of the input. This improves the previously best running time for Max-Bisection of [DBLP:journals/tcs/HanakaKS21] by a factor $t^2$. Under common hardness assumptions, neither the dependence on $t$ in the exponent nor the dependence on $n$ can be reduced [DBLP:journals/tcs/HanakaKS21,DBLP:journals/jcss/EibenLM21,DBLP:journals/talg/LokshtanovMS18]. Our framework can be applied to other cut problems like Min-Edge-Expansion, Sparsest-Cut, Densest-Cut, $\\beta$-Balanced-Min-Cut, and Min-Bisection. It also works in the setting with arbitrary weights and directed edges.","url_abs":"https://arxiv.org/abs/2101.00694v3","url_pdf":"https://arxiv.org/pdf/2101.00694v3.pdf","source":{"archive":"pwc-archive (Hugging Face), CC BY-SA 4.0","snapshot":"2025-07-28","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","row_kind":"links_only","authors_date_abstract":"arXiv metadata, CC0 1.0 (https://info.arxiv.org/help/license), from the Kaggle arXiv metadata snapshot of 2026-09-12"},"code_links":[{"paper_slug":"an-optimal-fpt-algorithm-parametrized-by","repo_url":"https://github.com/tim-we/td-bisection","is_official":0,"mentioned_in_paper":0,"mentioned_in_github":1,"framework":"none","reach":null}],"tasks":[],"methods":[],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"atlas_url":null,"mcp":null,"developers":"https://syntology.ai/developers"},"arxiv_metadata":null,"syntology_extracted_results":null}